Three-term inversion of prestack seismic data using a weighted l2, 1 mixed norm

Abstract

We present a new inversion method to estimate, from prestack seismic data, blocky P- and S-wave velocity and density images and the associated sparse reflectivity levels. The method uses the three-term Aki and Richards approximation to linearise the seismic inversion problem. To this end, we adopt a weighted mixed l2, 1-norm that promotes structured forms of sparsity, thus leading to blocky solutions in time. In addition, our algorithm incorporates a covariance or scale matrix to simultaneously constrain P- and S-wave velocities and density. This a priori information is obtained by nearby well-log data. We also include a term containing a low-frequency background model. The l2, 1 mixed norm leads to a convex objective function that can be minimised using proximal algorithms. In particular, we use the fast iterative shrinkage-thresholding algorithm. A key advantage of this algorithm is that it only requires matrix–vector multiplications and no direct matrix inversion. The latter makes our algorithm numerically stable, easy to apply, and economical in terms of computational cost. Tests on synthetic and field data show that the proposed method, contrarily to conventional l2- or l1-norm regularised solutions, is able to provide consistent blocky and/or sparse estimators of P- and S-wave velocities and density from a noisy and limited number of observations.